58 Jean-Marie De Koninck
x N(x)
10 10
100 33
1000 213
x N(x)
104
1 538
105
11 872
106
95 428
x N(x)
107
806 095
108
6 954 793
109
61 574 510
the smallest number n such that λ0(n) = λ0(n + 1) = . . . = λ0(n + 7) = 1,
where λ0 stands for the Liouville function; moreover, in this case, we also have
λ0(n + 8) = 1 (see the number 1 934 for the list of the smallest numbers at
which the λ0 function takes the value 1 at consecutive arguments).
215
the second solution of σ2(n) = σ2(n + 2) (see the number 1 079).
216
the only cube which can be written as the sum of the cubes of three consecutive
numbers: 216 = 63 = 33 + 43 + 53;
the tenth 3-powerful number (counting 1 as a 3-powerful number); if nk stands
for the kth 3-powerful number, then n10 = 216, n100 = 52 488, n1
000
=
25 153 757, n10
000
= 16 720 797 973, n100
000
= 13 346 039 198 336 and
n1
000 000
= 11 721 060 349 748 875.
217
the smallest number 1 for which the sum of its divisors is a fourth power:
σ(217) = 44; the sequence of numbers satisfying this property begins as follows:
217, 510, 642, 710, 742, 782, 795, 862, 935, . . . ; it is also the smallest number
n such that σ(n) is an eighth power: σ(217) = 28.
220
the number which, when paired with the number 284, forms the smallest ami-
cable pair: two numbers are called amicable if the sum of the proper divisors of
one of them equals the other: here σ(220) 220 = 284 and σ(284) 284 = 220.
221 (=
102
+
112
=
52
+
142)
the number used by Euler to show how to find its factors given that two distinct
representations of it as the sum of two
squares70
are known.
70Here is Euler’s method. Let n be an odd number which can be written as the sum of two squares
in two distinct ways:
n =
a2
+
b2
=
c2
+
d2,
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